Richmond Geometry Meeting
Knots, Moduli, and Strings
Virginia Commonwealth University
June 2 - 4, 2023
The Richmond Geometry Meeting will focus on emergent research topics while bringing together researchers in algebraic geometry, low-dimensional topology, and mathematical physics.
In summer 2023, we will highlight developments in moduli theory together with new perspectives in Floer homology and low-dimensional topology. This year's format will be in-person.
Organizers
We would appreciate your feedback: Anonymous Feedback for RGM Organizers.
"Chloe (Primary Title)" by Jaume Plensa, viewable at the Virginia Museum of Fine Arts in Richmond, VA. Photo by N. Tarasca.
Speakers
Panelists
Sudharshana Srinivasan (Altria Group)
Photo by Biji Wong.
Logistics
Location. All lectures and the Career Panel will be held in the Academic Learning Commons Room 1104.
Accommodation. A block of hotel rooms has been reserved at the Homewood Suite Downtown at a discounted rate. To take advantage of the discount, please mention the name of this event when booking.
Childcare. For participants wishing to arrange childcare in Richmond, VA, the Virginia Department of Social Services provides a child day care search tool here:
Please select the search criterion "Richmond City" or the nearby counties "Henrico" or "Chesterfield." A list of contact information for providers with some drop-in childcare options can also be found here. Other childcare options in Richmond include the YMCA, numerous area churches, and the Weinstein JCC. It may also be possible to locate home daycare providers or babysitters for hire through Care.com.
Poster Session
The meeting will feature a poster session with the aim of showcasing research by early-career participants.
Titles and abstracts of accepted posters will appear on the website at a later date.
Everyone is encouraged to apply to present their work at the poster session. The poster session application is an optional part of the registration.
Schedule
Displayed time is in EST
Day 1. Friday, June 2, 2023
12:45 PM – 01:00 PM Welcome message
01:00 PM – 02:00 PM Slava Krushkal
03:00 PM – 04:00 PM Beibei Liu
05:00 PM – 06:00 PM Charles Doran
Day 2. Saturday, June 3, 2023
09:00 AM – 10:00 AM Hülya Argüz
11:00 AM – 12:00 PM Reimundo Heluani
02:00 PM – 03:00 PM Career Panel
03:30 PM – 04:30 PM Jim Bryan
05:00 PM – 06:30 PM Poster Session in LSEE Building
07:45 PM Social Event at Brambly Park
Day 3. Sunday, June 4, 2023
09:00 AM – 10:00 AM Antonio Alfieri
10:30 AM – 11:30 AM Dragos Oprea
12:00 PM – 01:00 PM Pavel Putrov
Please note the nearby Workshop on New Developments in 3- and 4-Manifold Topology at the University of Virginia happening June 5-7, 2023. Charlottesville is just a short drive from Richmond, VA.
Lecture Abstracts
Antonio Alfieri
Instanton Floer homology of almost-rational plumbings
Plumbed three-manifolds are those three-manifolds that can be realized as links of isolated complex surface singularities. Inspired by Heegaard Floer theory Nemethi introduced a combinatorial invariant of complex surface singularities (lattice cohomology) that was recently proved to be isomorphic to Heegaard Floer homology (Zemke). I will expose some work in collaboration with John Baldwin, Irving Dai, and Steven Sivek showing that the lattice cohomology of an almost-rational singularity is isomorphic to the framed Instanton Floer homology of its link. The proof goes through lattice cohomology and makes use of the decomposition along characteristic vectors of the instanton cobordism maps found by Baldwin and Sivek.
Hülya Argüz
Quivers, flow trees and log curves
Donaldson-Thomas (DT) invariants of a quiver with potential can be expressed in terms of simpler attractor DT invariants by a universal formula. The coefficients in this formula are calculated combinatorially using attractor flow trees. In joint work with Bousseau (arXiv:2302.02068), we prove that these coefficients are genus 0 log Gromov-Witten invariants of d-dimensional toric varieties, where d is the number of vertices of the quiver. This result follows from a log-tropical correspondence theorem which relates (d-2)-dimensional families of tropical curves obtained as universal deformations of attractor flow trees, and rational log curves in toric varieties.
Jim Bryan
The enumerative geometry of nano banana manifolds
The Hodge numbers of a Calabi-Yau threefold X are determined by the two numbers h^{1,1}(X) and h^{1,2}(X) which can be interpreted respectively as the dimensions of the space of Kahler forms and complex deformations respectively. We construct examples of rigid Calabi-Yaus (h^{2,1}=0) with Picard number 4 (h^{1,1}=4). These manifolds are of “banana type” and have among the smallest known values for Calabi-Yau Hodge numbers. We (partially) compute the partition functions of these manifolds and in particular show that the genus g Gromov-Witten potential is given by a weight 2g-2 Siegel paramodular form. We will explain the construction and explain why manifolds of “banana type” are amenable to computing enumerative invariants. This is joint work with Stephen Pietromonaco.
Charles Doran
Motivic Geometry of Two-Loop Feynman Integrals
We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into a mixed Tate piece and a variation of Hodge structure from families of hyperelliptic curves, elliptic curves, or rational curves depending on the space-time dimension. We give more precise results for two-loop graphs with a small number of edges. In particular, we recover a result of Spencer Bloch that in the well-known double box example there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the “non-planar” two-loop tardigrade graph is that of a family of K3 surfaces of generic Picard number 11. Lastly, we show that generic members of the multi-scoop ice cream cone family of graph hypersurfaces correspond to pairs of multi-loop sunset Calabi-Yau varieties. Our geometric realization of these motives permits us in many cases to derive in full the homogeneous differential operators for the corresponding Feynman integrals.
Reimundo Heluani
PBW bases of Ising modules
We describe PBW bases of the unique three irreducible modules of the Virasoro Lie algebra with central charge c=1/2. We use these bases to find new bi-variable character formulas for these modules and describe new Rogers-Ramanujan-type identities from them. This is a report on the thesis of Diego Salazar Gutierrez (IMPA).
Slava Krushkal
A 4-manifold invariant from topological modular forms
I will discuss work in progress, joint with Sergei Gukov, Lennart Meier, and Du Pei, concerning a construction of a 4-manifold invariant using the theory of topological modular forms, and TQFT properties of this invariant. This is a mathematical construction related to a particular instance of the Gukov-Pei-Putrov-Vafa program associating an invariant of 4-manifolds to certain 6-dimensional superconformal field theories.
Beibei Liu
Skein exact sequence in Heegaard Floer homology
Skein exact sequences for links show up in Khovanov homology and various Floer homologies. In this talk, we will talk about the skein exact sequence for links from the surgery exact triangle in Heegaard Floer homology. As an application, this can be used to study splitting numbers and splitting maps for links. In particular, we do the explicit computation for the split maps of the torus link T(n, n) and compare it with the computation in the deformed HOMFLY homology.
Dragos Oprea
Cycles on the moduli space of abelian varieties
I will present a few new results and conjectures regarding tautological classes on the moduli space of principally polarized abelian varieties. The case of abelian 6-folds is particularly interesting. This is based on joint work with Samir Canning and Rahul Pandharipande.
Pavel Putrov
Analytically continued Chern-Simons theory on plumbed 3-manifolds
I will present a finite-dimensional model for analytically continued Chern-Simons theory on closed 3-manifolds that are described by plumbing trees. From this model, one can define a collection of topological invariants labeled by pairs of flat connections and valued in formal power series with integral coefficients. I will also comment on a possible categorification, which can be interpreted as a finite-dimensional model of Fukaya-Seidel category of Chern-Simons functional on the space of SL(2,C) connections.
Poster Abstracts
1. Yifeng Huang (University of British Columbia)
Motive of the punctual Quot schemes on singular curves
(Joint with Ruofan Jiang) The Quot scheme of degree-n rank-0 quotients of the trivial vector bundle O^d is a generalization of the Hilbert scheme of points; the latter is when d=1. We study the motive of these Quot schemes on the cusp singular curve using Gröbner stratification, proving a rationality result that extends the known one for Hilbert schemes of any singular curve. We give explicit computations in d<=3, which shows strong conjectural patterns that will lead to a formula for general d. These patterns include a functional equation, which extends the known one for Hilbert schemes of any Gorenstein curve.
2. Swan Klein (George Mason University)
Combinatorial Formulas for the Equivariant Cohomology of Peterson Varieties
Our goal was to verify a conjecture about the decomposition of the restriction of Schubert classes associated with transpositions to the Peterson variety into a linear combination of Peterson classes. Using a corollary of the AJS/Billey formula, we reduced the conjecture to a more concise combinatorial question about counting reduced words for transpositions embedded into long words. We uncovered an elegant visual framework for understanding these combinatorial questions and proved our conjecture in a specific subcase. With future work, we hope to prove the remaining cases of the conjecture and extend our combinatorial strategy to as many types of Schubert classes as possible.
3. Louisa Liles (University of Virginia)
Z-double-hat, Brieskorn spheres, and (quantum) modularity
Z-hat is a q-series invariant of negative definite plumbed 3-manifolds. It has been extended to the (q,t)-series Z-double-hat, which specializes to Z-hat when t=1. It is known that Z-hat of a Brieskorn sphere is a quantum modular form. We calculate Z-double-hat of Brieskorn spheres, and show that when t is any root of unity, the resulting q-series is a finite sum of modular and quantum modular forms.
4. Jae Hwang Lee (Colorado State University)
A Quantum H^∗(G)-module via Quasimap Invariants
For X a smooth variety or Deligne-Mumford stack, the quantum cohomology ring QH∗(X) is a deformation of the usual cohomology ring H∗(X), where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov-Witten invariants. For a GIT quotient V //G, the cohomology ring H∗(V //G) also has the structure of a H∗(G)-module. In this work, we use quasimap invariants with light points and a modified version of the WDVV equation to define a quantum deformation of this H∗(G)-module structure. Using localization, we explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the Batyrev ring when the target is a semipositive toric variety.
5. Yangxiao Luo and Shunyu Wan (University of Virginia)
Oriented Thompson links and grid diagram
Given any oriented Thompson link which is represented by certain pair of binary trees with same leaf number, we construct a canonical grid diagram associated to it. Using the grid representation we show that the parity of leaf number is the same as the parity of number of link components. This is joint work.
6. Rob McConkey (Michigan State University)
Linear Bounds on the Cross Cap Number of Links
The cross-cap number of a link is an invariant which considers the non-orientable spanning surfaces of the link, similar to how the genus of a link depends on the orientable spanning surfaces. In 2014 Kalfagianni and Lee found linear bounds for the cross-cap number of alternating links in relation to the coefficients of the Jones Polynomial. But what happens when we begin to look beyond alternating links? We consider a couple of families of links where such linear bounds cannot be found. Then talk about a family where we can find linear bounds for the cross cap number with respect to the twist number.
7. Leonard Mushunje (Columbia University)
High Dimensional Functional Data Analysis via Algebraic Geometry
When regressing high-dimensional functional data, challenges are often encountered mainly for the in-sample than out-sample results and even worse when subjected to the curse of dimensionality. For example, on the in-sample, minimal bounds on the eigenvalues of the covariance matrix for the covariates, when using ridge regression, are not generally considered. This study aims to explore the in-sample MSPE properties of different regression methods (except ridge regression) and understand whether the eigenvalue lower bounding conditions are generally avoidable in high-dimensional Hilbert settings.
8. Greyson Potter (Boston University)
Non-perturbative topological recursion and SL(2,C) Chern-Simons Theory
I will discuss the conjectured relationship between topological recursion and Chern-Simons theory with complex gauge group SL(2,C), known as the generalized volume conjecture (GVC) for hyperbolic knots. The conjecture states that there is asymptotic agreement between three generating functions: the colored Jones polynomials of the knot, the partition function of Chern-Simons theory with complex gauge group SL(2,C) associated to the knot complement, and the non-perturbative wave-function arising from topological recursion on the A-polynomial of the knot. I will also discuss a new algorithm for computing topological recursion via higher quantum Airy structures that uses graph sums and an efficient graph generation algorithm, which was used to verify the GVC to higher order than was previously accessible in several cases.
9. Tonie Scroggin (UC Davis)
Computing (Co)homology on Two Strand Braid Varieties Using Cluster Algebras
We define the braid variety as an link invariant. The homology of a braid variety is related to the Khovanov-Rozansky homology of a corresponding link, which is notoriously difficult to compute. The braid variety is isomorphic to a positroid variety; therefore, the braid variety has a cluster structure. Any cluster structure has a canonical 2-form with constant coefficients in all cluster charts, which gives an interesting class in de Rham cohomology of degree 2, and an interesting operator in link homology. Using the cluster structure, we compute the homology of the braid variety using the De Rham cohomology.
10. Alec Traaseth (University of Virginia)
Combination Theorems for Discrete Convergence Groups
The classical Klein-Maskit combination theorems give a way to construct new Kleinian groups out of simpler ones. In joint work with Teddy Weisman, we prove versions of these theorems in the setting of discrete convergence groups, a far reaching generalization which includes isometry groups of any Gromov-hyperbolic metric space.
11. Weihong Xu (Virginia Tech)
Quantum K-theory of incidence varieties
We prove a conjecture of Buch and Mihalcea in the case of the incidence variety X=Fl(1, n-1, n) and determine the structure of its (T-equivariant) quantum K ring. In particular, we derive a positive Chevalley formula and a presentation for the equivariant quantum K ring of X, as well as a positive Littlewood--Richardson rule for the non-equivariant quantum K ring of X. Our proof is via the study of rationality properties of certain Gromov--Witten varieties, which are subvarieties of the Kontsevich moduli space of 3-pointed genus 0 stable maps to X. We also conjecture a presentation for the quantum K ring of any flag variety Fl(r_1, ..., r_k, n). Part of this poster is based on joint work with W. GU, L. Mihalcea, E. Sharpe, and H. Zhang.
12. Ming Zhang (UC San Diego)
Quantum K-theory and Quot Scheme
Quantum K-theory of Grassmannians was first studied by Buch and Mihalcea, and it has rich combinatorial structures. They used the ``quantum to classical principle'' and reduced the computation of three-pointed quantum K-invariants to that over a two-step flag manifold. In this poster, I will present a different approach to studying quantum K-invariants of Grassmannians using K-theoretic Quot scheme invariants. This is joint work in progress with Shubham Sinha.
13. Shuo Zhang (University of Minnesota Twin Cities)
Lagrangian Clean intersection
Lagrangian surgery is a basic operation to obtain new Lagrangian submanifolds, discovered by Polterovich. It has been known that in Floer theory, Lagrangian surgery at a transversal intersection corresponds to the mapping cones of the morphism represented by the resolved intersections. If the Lagrangian submanifolds intersects cleanly, the expected operation in Fukaya category among the experts is the mapping cone of the morphism represented by the fundamental class of the clean intersection. In a joint work with Mak and Wu we'll prove this for exact Lagrangians.